Question

In Exercises \(35-38,\) use Taylor's Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. $$ e \approx 1+1+\frac{1^{2}}{2 !}+\frac{1^{3}}{3 !}+\frac{1^{4}}{4 !}+\frac{1^{5}}{5 !} $$

Short answer

The approximate error would be less than or equal to \(\frac{e}{6!}\). The actual error is given by the absolute value of the difference between the actual value of \(e\) and the approximated value. These are compared to validate the Taylor theorem error bound.

Step-by-step solution

Calculate the Taylor series approximation

The approximation for \(e\) using Taylor series up to the fifth term is: \(e \approx 1+1+\frac{1^{2}}{2 !}+\frac{1^{3}}{3 !}+\frac{1^{4}}{4 !}+\frac{1^{5}}{5 !}\). Calculate this value for further error analysis.

Compute the upper bound of the error using Taylor's Theorem

The remainder term \(R_n(x)\) according to Taylor's theorem is given by \( \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}\). Here, as the function is \(e^x\), \(f^{(n+1)}(c)\) equals to \(e^c\) and, \(x=a=1\). So, the error is bounded by \(\frac{e}{(n+1)!}\). It is required to use the next term \(n=6\) in the series to get the error which would be \(\frac{e}{6!}\). So calculate this error bound.

Calculate the actual error

The actual error is given by the difference between the actual value and the approximated value, i.e., \(|e - approximation|\). Calculate the actual value of \(e\) using standard methods and then find the difference with the approximation from step 1.

Compare the approximate and exact error

Compare the approximate error obtained in step 2 with the actual error derived in step 3. This step verifies the correctness of the error upper bound estimation.